Computational Analysis of Quantitative Characteristics of Some Residual Properties of Solvable Baumslag–Solitar Groups

Let Gk be defined as Gk = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\langle {a,b;{{a}^{{ - 1}}}ba = {{b}^{k}}} \right\rangle $$\end{document}, where k ≠ 0. It is known that if p is a prime number then Gk is residually a finite p-group iff p|k – 1. It is also known that if p and q are primes not dividing k – 1, p < q and π = {p, q} then Gk is residually a finite π-group iff (k, q) = 1, p|q – 1 and the order of k in the multiplicative group of the field ℤq is a p-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let fk(x) be the number of sets {p, q} such that p < q, p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nmid $$\end{document}k – 1, q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nmid $$\end{document}k – 1, (k, q) = 1, p|q – 1, the order of k modulo q is a p-number, and p and q are chosen among the first x primes. We state that, if 2 ≤ |k| ≤ 10 000 and 1 ≤ x ≤ 50 000, then for almost all considered k the function fk(x) can be approximated quite accurately by the function αkx0.85, where the coefficient αk is different for each k and {αk|2 ≤ |k| ≤ 10 000} ⊆ (0.28; 0.31]. In addition, the dependence of the value fk(50 000) on k is investigated and an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion is proposed. The results may have applications in the theory of computational complexity and algebraic cryptography.

1. SOME APPROXIMATION PROPERTIES OF THE CONSIDERED GROUPS Following [1], for each nonzero integer number k by G k we denote a group that is given by the presentation G k = . It belongs to the well-known family of Baumslag-Solitar groups BS(m, n) = , where m and n are nonzero integer numbers. G. Baumslag and D. Solitar began studying the residual properties of the G k group in [2]. We recall that a group G is called residually a #-group for a class of groups # if for any non-trivial element g ∈ G there exists a homomorphism ϕ of the group G onto a group from the class # such that gϕ ≠ 1. In [2,3] it was established that a group G k is residually finite, that is, is residually a finite group. Investigation of the residual finiteness of this group was continued by D.I. Moldavanskii. In [4] he proved the following criterion for the group G k to be residually p-finite, where p is a prime number.
Theorem 1 [4,Theorem 3]. We suppose that p is a prime number. A group G k is residually p-finite iff p divides k -1.
In [5] Ivanova and Moldavanskii considered the residual π-finiteness of the group G k , where π is some set of prime numbers. We recall that an integer number is a π-number if all its prime divisors belong to the set π and a periodic group is called a π-group if the orders of all its elements are π-numbers. Theorem 2 given below provides the criterion for the residual π-finiteness of a group G k for an arbitrary set π of prime numbers. We clarify that in the formulation of Theorem and everywhere below the order of an integer number x modulo integer number y is the order of the number x in the multiplicative group of the ring Z y .
Theorem 2 [5,Theorem 1]. We suppose that π is an arbitrary set of prime numbers. The group G k is residually π-finite iff there exists a π-number s > 1 that is coprime with k whose order modulo number k also is a π-number. The criterion contained in Theorem 2 is not constructive, that is, it provides no algorithm that allows, for a given number k and a given set π, determining whether a group G k is residually π-finite. Such an algorithm exists for a two-element set of prime numbers and is provided by the next Theorem 3. We note that in this theorem we consider only pairs of prime numbers p and q, each of which does not divide k -1, because in the opposite case the group G k is residually p-finite or residually q-finite, according to Theorem 1.
Theorem 3 [5,Theorem 2]. We suppose that p and q are prime numbers that satisfy the conditions listed below: (a) p < q.
(3) The order of k modulo q is a p-number.
Although Theorem 3 has an effectively determined condition, it provides no answer to the question of whether, for any value of k, a pair of prime numbers (p, q) exists that satisfies its condition. It is not hard to prove that at k = ±1 no such pair exists. At |k| > 1 the existence of a pair of prime numbers with the required properties is established by Theorem 4 [5,Theorem 3]. If |k| > 1, then, for any prime number p not dividing k -1, there exists a prime number q not dividing k -1 and such that the group G k is residually π-finite, where π = {p, q}.
The question about the number of two-element sets of prime numbers that satisfy the conditions of Theorem 3 is left open by Theorem 4. In the current work we study this question under the condition that 2 ≤ |k| ≤ 10000 and the elements of the set are chosen from the first 50000 prime numbers. These restrictions are caused by the execution time of the program counting two-element sets of prime numbers and selecting those of them for which the conditions of Theorem 3 are valid. An attempt at a further increase in the range of considered prime numbers leads to incomparable growth in the time of calculating the initial data for performing our study; we go into the details about this in Section 4.

THE DEPENDENCE OF THE NUMBER OF TWO-ELEMENT SETS
THAT SATISFY THE CONDITIONS OF THEOREM 3 ON k We suppose that |k| > 1 and n ≥ 1. We use the following denotations: 3(n) is the set composed of the first n prime numbers. 6 k (n) is the set of pairs of prime numbers (p, q) that satisfy the conditions (a)-(c), 1-3 of Theorem 3 and such that p, q ∈ 3(n). f k (n) is the cardinality of the set 6 k (n).
We also set Upon direct computation of elements of the sets 6 k (50000) for 2 ≤ |k| ≤ 10000, we established that the values of C(k) vary in a considerably wide ranges. If we consider the function C : _ → Z as a random variable, then, under the assumptions of equal probability of all elementary events, we have We may extract some values of k for which the value C(k) is substantially different from M(C). The inequality C(k) < 1/2M(C) is valid for 76 elements of the set _, and all these numbers are negative. Among them we should note the values of k for which C(k) < 1/4M(C): -256; -16; -6561; -4096; -576; -9216; and -2916. All positive values of k ∈ _ satisfy the relation C(k) > 0.73M(C).
In the case where the value of C(k) exceeds the average value M(C), the picture is the opposite. The inequality C(k) > 2M(C) is fulfilled for 37 values of k, and all these values are positive. We note k = 4096, because C(4096) = 11878 > 4M(C) and C(k) < 3M(C) for all other k ∈ _. All negative values of k ∈ _ satisfy the relation C(k) < 1.62M(C).
The points at which the function C(k) has the 20 largest and 20 smallest values are given in Table 1. By analyzing its content, it is logical to assume that the value of the function C(k) is substantially associated with the decomposition of the number k into prime multipliers. Studying this dependence should be the subject of further research.
There is one more natural question: can we manually find a pair of prime numbers that satisfy the conditions of Theorem 3 relatively quickly. To answer this question, we investigated the function c(k) corresponding to the case where prime numbers are chosen from the set 3(4) = {2, 3, 5, 7}.
We established that for 10 000 of 19 998 considered values of k the inequality c(k) > 0 is true; and among the 10 000 values of k there are almost equal shares of positive and negative ones: 4998 positive and 5002 negative values. Thus, for approximately half of the elements of the set _ the sought pair of prime numbers can be rapidly found by an exhaustive search. In addition, we established that c(k) ≤ 3 for any k ∈ _. The numbers of the elements of the set _ for which the function c(k) has the values 0, 1, 2, and 3 are given in Table 2.

THE DEPENDENCE OF THE NUMBER OF TWO-ELEMENT SETS THAT SATISFY THE CONDITIONS OF THEOREM 3 ON THE NUMBER OF CONSIDERED PRIMES
In this section we investigate the question of how the number of pairs of prime numbers that satisfy the conditions of Theorem 3, that is, the value of f k (n) vary dependent on the number n of considered prime numbers at a fixed k. Can we approximate this dependence by one of the elementary functions well? From the results of the previous section, it is clear that for different values of k the functions f k (x) have substantially different values. Does the value of k affect the character of the dependence or are the functions f k (x) different from each other only by some multiplier?  To answer to these questions, we visualized the computed values of the functions f k (x) for certain values of k (see Fig. 1). As a result, the hypothesis occurs that for different values of k the functions f k (x) have the same character of the dependence similar to the logarithmic or power one (with an exponent below 1). The first attempts of approximations showed that a logarithmic function does not fit. Therefore, we decided to seek the approximate function in the form g k (x) = αx β , where the coefficients α and β are, in general, dependent on k.
For a given k and β the search for the multiplier α was performed using the condition where which leads to We determined β in two ways.

Method 1.
An exhaustive search for each k ∈ _ of all values from the set @ containing the numbers from 0.7 to 0.99 with a step 0.01. The minimum element of the set @ was found during the preliminary calculations with a lower number of points for which we computed the functions f k (x).
For a fixed k ∈ _, for each b ∈ @ we determined the corresponding values after which we chose the element b ∈ @ for which the value δ(b) was at a minimum. This element is further denoted by β k .

Method 2.
Using the same element b ∈ @ for all k ∈ _. We chose an element b ∈ @ that was closest to the average value of the numbers β k (k ∈ _) obtained by the first method. This element is further denoted by β avg .
We note that the determination of β with higher accuracy was not conducted, because the original problem consisted of describing the general form of the functions f k (x) and not in the obtaining the pos-2 50000 1 1 m i n , ( ) x k  The possibility of choosing the coefficient β by the second method is provided by the peculiarity of the distribution of the numbers β k (k ∈ _). We suppose that the functions β : _ → @ and η : @ → Z are determined as follows: The values of the function η presented in Table 3 indicate that if the function β is considered to be a discrete random variable, then it has the normal distribution. Let us check the satisfaction of the three-sigma rule for it. We have M(β) ≈ 0.8531 and σ(β) ≈ 0.0206. The random value β(k) is contained in the range (M(β) -3σ(β); M(β) + 3σ(β)) with the probability ≈0.9954, in the range (M(β) -2σ(β); M(β) + 2σ(β)) with the probability ≈0.9531, and in the range (M(β) -σ(β); M(β) + σ(β)) with the probability ≈0.6688. The three-sigma rule is not satisfied, but the obtained probabilities are slightly different than the necessary values; therefore, we can state that the distribution of the random value β(k) is close to the normal one.
We note that the two smallest values of the function β correspond to the elements of the set _ from Table 1. Namely, β(k) = 0.71 at k = -256 and β(k) = 0.75 at k = -2916. Concerning the maximum of the function β(k) = 0.94, the situation is different: in five of six points where it was reached the random variable C has a value close to M(C). The exception is k = -1849 for which C(k) = 1630. Thus, we observe no pronounced relationship between the points at which the functions C and β have values close to the limit ones.
In using methods 1 and 2 for each k ∈ _, we found two values of the coefficient α; we denote them by and , respectively, and note that For each k ∈ _ estimating the quality of approximation of the function f k (x) was carried out by determining the number that maximizes the values for at least 95% of the elements of the set 3(50000). We say that a number k ∈ _ possesses the property P(r), where r ∈ (0; 1) if or, which is the same, 0.064;0.775 , 0.028;1.186 . In Table 4 we present the information about the numbers of elements of the set _ that do not possess the property P(r), dependent on the method of computing the coefficient β. As anticipated, the application of the first method gives a better approximation. However, when the second method is used, the majority of numbers from the set _ has the property P(0.15), which can also be considered a satisfactory result.
It is natural to assume that if for certain k ∈ _ the value β is substantially different from β avg , then the function g k (x) = approximates the function f k (x), poorly; therefore, the property P(r) is satisfied for the number k just for a sufficiently large r. Nevertheless, we revealed no correspondence between the value |β k -β avg | and the minimum r ∈ (0; 1) for which k has the property P(r).
Thus, the analysis we performed supports the following conclusion.

ON THE COMPUTATIONAL COMPLEXITY OF CHECKING
THE CONDITIONS OF THEOREM 3 All the above described results were obtained using the program we implemented in C++ programming language (GCC) with the use of the C RTL and C++ STL. From the performed computations, the highest computational complexity was associated with the search for the elements of the set 6 k (n). To conduct this operation, we need to check the satisfaction of the conditions of Theorem 3 for n(n -1)/2 pairs (p, q), where p < q. In the course of each test, we need to prove that (1) p does not divide k -1.
(3) (k, q) = 1 (which, in view of the prime character of the number q, is equivalent to the condition q k).  We recall that the order of k modulo q is the lowest number x from the set {0, 1, ..., q -1} that satisfy the relationship k x ≡ 1 (mod q). Thus, determining the order of number k is simply the computation of the discrete logarithm in the multiplicative group of the field Z q . As is well-known, this problem is rather complex: the widespread algorithms of its solution have a complexity on the order of O( ).
To make the algorithm more efficient, we note that in reality we do not need to compute the order s of the number k, but only need to determine whether it is a p-number. We suppose that m is the maximum integer number that satisfies the relationship p m |q -1. Because the considered group has the order q -1, s|q -1; therefore, the order s is a p-number iff it coincides with one of the numbers 1, p, p 2 , ..., p m . Moreover, s ≠ 1, because q does not divide k -1.
Thus, to satisfy the last, fifth, operation, we need m ≤ logp(q -1) trials. Each of them requires raising either the number k (for the first trial) or the number considered in the previous trial to the power p. When we use the binary algorithm for raising to a power (see, for instance, [6, Section 1.2]), we need approximately O(,) multiplications for this purpose, where , = + 1 is the length of the binary representation of the number p. In total, the proposed algorithm for testing the pair (p, q) for its correspondence to the conditions of Theorem 3 has the complexity which is lower than that in computing the discrete logarithm.

CONFLICT OF INTEREST
The author declares that she has no conflicts of interest.